Question

Let p(x) be a quadratic polynomial with real co-efficients such that p(11) = 181, and x² - 2x + 2 ≤ p(x) = 2x² - 4x + 3 for any real number x. Find the value of p(6).​

Answer 1

Answer:

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Answer 2

Let p(x) be a quadratic polynomial with real co-efficients such that p(11) = 181, and x² - 2x + 2 ≤ p(x) = 2x² - 4x + 3 for any real number x.

Find the value of p(6).

Let Q(x) = P(x) - (x² - 2x + 2), then 0 ≤ Q(x) ≤ (x-1)² (note this is derived from the given inequality chain).

Therefore, $0 ≤ Q(x+1) ≤ x²

→ Q(x+1) = Ax² for some real value A.

$Q(11) = 10²A

→ P(11)-(11²-22+2) = 100A

→ 80 = 100A

→ A = 45

Q(16)=15²A=180

→ P(16)-(16²-32+2) = 180

→ P(16)=180+226 = 406